Dynamics of Hodgkin and Huxley Model with Conductance based Synaptic Input

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1 Proceedings of International Joint Conference on Neural Networks, Dallas, Texas, USA, August 4-9, 2013 Dynamics of Hodgkin and Huxley Model with Conductance based Synaptic Input Priyanka Bajaj and Akhil Ranjan Garg Abstract --The original Hodgkin and Huxley equations are landmark equations explaining the generation of action potential in a biological neuron. Moreover, many studies have been done on the Hodgkin and Huxley model with constant injected current. Here we present an Extended Hodgkin and Huxley model with conductance based excitatory and inhibitory synaptic inputs. It is asserted that the Hodgkin and Huxley model remains robust with the all kinds of synaptic inputs. Moreover, this model is more tractable to a biological neuron. I. INTRODUCTION EURON is the basic functioning unit of the brain. It Nforms a complex network with help of synapses and interconnections with other neurons. Each synapse is a channel with which a neuron communicates with other neuron. There are about neurons in human brain, communicating with each other through synapse. This neural network is capable of performing complex computational task and process information [1]. Many modeling studies have been carried out proposing various models of neurons. While modeling a neuron a delicate compromise is often made between a detailed model that captures all the known features of neuron and over simplified model, which captures some basic features of a neuron [2]. As making a detailed model sometimes leads to a condition where it becomes difficult or rather impossible to have a control over what is going on. On the other hand, over simplified model may show the behavior or properties of a model neuron which do not exist in realistic biological neuron. Since the introduction of first Integrate and Fire neuron model by Lapicque in 1907 [3], the basic IF model and its variants are still considered as conceptually simple and analytically tractable model of neuron. It is most widely used model for exploring general principles and characteristic of realistic biological neuron. However, these models are a low level compromise between biological system and its equivalent mathematical model. One of the major drawbacks of basic IF model and its variants is that in these models generation of spikes uses manually implemented reset mechanism, which is far off from the real mechanism of generation of action potential [4]. Manuscript received March 1, This work was supported in part by Ministry of Science and Technology, Department of Science and Technology under grant number SR/CSI/26/2008 to Akhil R. Garg. Priyanka Bajaj is M.E scholar with the Department of Electrical Engineering, Faculty of Engineering and Architecture, J.N.V University, Jodhpur, India ( bajajpriyanka05@gmail.com) Akhil Ranjan Garg is with Department of Electrical Engineering, Faculty of Engineering and Architecture, J.N.V University, Jodhpur, India ( garg_akhil@yahoo.com) In 1952, Hodgkin and Huxley [5] proposed a model for generation of action potential, which includes a set of nonlinear ordinary differential equation that captures the biological mechanism of generation of action potential. In their work and other models based on their seminal work it was shown that with the injection of constant applied current there is a generation of action potential. We present Extended Hodgkin and Huxley model that incorporates conductance based synaptic inputs in addition to constant injected current. The voltage-dependent conductance in each neuron determines its excitability and the way it responds to external drive. Further, fundamental mechanism by which neurons maintains appropriate response levels to the changing features of their synaptic inputs is often characterized by the neuronal input-output relationship for varying excitatory and inhibitory synaptic input [6],[7]. Therefore, it was of interest to obtain such neuronal input-output relationships for varying excitatory and inhibitory synaptic input. We also obtained the nature of action potential and frequency - current response of the neuron depending on the Poisson distributed excitatory and inhibitory inputs. This study tests the robustness of the Hodgkin and Huxley model in presence of widely varying excitatory and inhibitory inputs. II. METHOD AND MATERIAL The overall simulation was performed in MATLAB. The extended Hodgkin and Huxley model was made as per the set of ordinary differential equations as described below. We examined the response characteristic of an Extended Hodgkin Huxley model as shown in fig.1 where we have replaced the constant injected current by conductance based excitatory and inhibitory synaptic inputs. In this we have modeled a network with N = 1000 excitatory connections and 100 inhibitory connections. Here, poisson distributed spike of various rates activates the excitatory synapse. The model also receives 100 inhibitory connections activated by Poisson distribution of variable rates. The membrane potential of the Extended Hodgkin and Huxley model neuron we used is determined by the ionic current equation as follows: _ _,,, /13/$ IEEE 1370

2 .. ;... ;... ; 1 ; 1 ; 1; ; ; ; Here I ext is the external constant injected current ( ). In addition, the net synaptic conductance of 1000 excitatory and 100 inhibitory synaptic connections are given by _ and g_ and are measured in units of leakage conductance of the neuron and are thus dimensionless. On arrival of presynaptic action potential at excitatory synapse _ (t) = _ (t) + g ex, and when the pre synaptic action potential arrives at inhibitory synapse _ (t) = _ (t) + g in, where g ex and g in are the peak synaptic conductances. Otherwise these conductance decay exponentially [8] as, _ _ and _ _ We have taken and as the time constants with the value 2 ms, g_exh = 0.015, g_inh = Presynaptic action potentials are poisson distributed of a particular excitatory and inhibitory frequency respectively. We simulated the model at range of constant injected current as 10 to 40, in addition to constant injected current of 7.5 synaptic excitatory input of pre synaptic firing rate varying from 5 Hz to 40 Hz was applied, (c) in addition to the input applied in a inhibitory input of pre synaptic firing rate varying from 5 Hz to 40 Hz was applied. The model was run for different input for simulation time of 5 sec with a time step of 0.05 ms. For these different inputs (a, b, c) we recorded the firing frequency, pattern of action potential generation, curve for m-n-h with respect to time and voltage and inter spike interval histogram of the Extended Hodgkin and Huxley model. III. PHYSIOLOGICAL DYNAMICS UNDERLYING HODGKIN AND HUXLEY MODEL Fig 1. Electrical Equivalent of Original Hodgkin and Huxley model with constant current injected. Electrical Equivalent of extended Hodgkin and Huxley model where constant injected current is replaced by conductance based excitatory and inhibitory synaptic inputs. Hodgkin and Huxley discovered existence of gated ion channels and modeled them as voltage dependent conductance namely sodium and potassium channels, also known as transient and persistent channels [9]. It was proposed that the transient channel consisted two gates m, which was the activation gate and h the inactivation gate. Whereas, the persistent channel contained only one gate n, the activation gate. Accordingly, sodium channel was proposed to have three states: (1) resting state (m gate is closed, h is open), (2) active state (m and h gates are open) and (3) inactive (m and h gates are closed) [10]. Whereas, potassium channel has only two states active and inactive only one gate governs it. The kinetics of opening and closing of gates of both channels is formalized as first order differential equation with opening and closing rate constants and for all gates. Moreover, this formulation precisely models the biological real action potential generation. With a stimulus, the depolarization of membrane potential occurs which rapidly opens the sodium active gate, m gate allowing the sodium ion to enter the membrane. This can be signified with the steep rise in the m curve as the external current depolarizes the membrane potential. Along with this, 1371

3 the potassium channel responds to the stimulus and there is opening of n gate, which is signified by increase of n values. However, with prolonged depolarization, the m gate begins to close and simultaneously there is opening of h gate, which is prominent through the m, n and h curves. Once the m gate closes, the channel cannot open with depolarization until it goes to resting state. Also, it is seen that there is rapid gating of m gate as compared to n and h because the time constant of m gate is much smaller than n and h. That is there is a steep rise and fall of m curve while n and h give a slow response. similar for low and high values of injected current. However, the rate of generation of action potential termed as firing rate increases linearly with increase in injected current. Here the firing rate was calculated as the inverse of the mean of inter spike interval [11]. IV. RESULTS 1) Simulation with constant current. We increased the constant injected current with a step size of 5. We simulated the model over wide range of constant input. It was observed that a substantial amount of external current was needed for the neuron to excite and generate action potential. In our model it was detected as 8.9. Gating behavior: The studies shows that the process of activation of sodium channel with the opening of m gate remains unchanged with the increase of constant excitation current as shown in fig 2 and. It can be seen that the opening and closing is as steep and same in both the cases. Whereas, the inactivation of sodium channel and closing of potassium channel becomes partial with increase in excitation. Fig.3. Voltage traces from the neuron model low injected current of 10 and high injected current 40. c) Inter spike Interval: With the increase in the excitation current there is a linear increase in the firing rate (fig 4), which decrease the inter spike interval. Fig.4 gives a comparative study of firing rate and ISI. The values of firing rate and ISI are normalized to 0-1 and the curve is fit across the increased external current. d) Dynamics of sodium and potassium conductances: Conductances of the model are voltage and time dependent. The major observation for sodium conductance was that with increase in the excitation, the sodium channel inactivates partially as seen on fig 5 and. In addition, the resting period of this channel reduces. Because of this, the sodium channels are not capable to activate properly with the depolarization of membrane potential. That is the level of conductance of sodium channel decreases while this affect is the potassium channel upto very small extent. Fig.2. Gating behavior of sodium and potassium channels with the increase of constant injected current. shows the variation in m,n and h with injected current 10 shows the variation in,n and h with the injected current The variation in m,n and h are shown in blue, green and red. b) Pattern of action potential and Spiking frequency:as seen in fig 3 and, the action potential is regular and Fig.4 Frequency current curve showing enhanced firing rate with increase in the constant injected current (shown in blue). The red curve indicated the Inter spike interval, which reduces with increase in the firing rate. 1372

4 Fig 5. Sodium and Potassium conductances variation with constant injected current. Shows enhanced Sodium conductance, which widens the action potential when injected current is 10. for constant injected current of 40 due to partial closure of sodium channel, channel does not open up to full extent. Red curve shows the sodium conductance and green shows potassium conductance. Fig 6 Gating behavior of sodium and potassium channels with synaptic excitatory input with presynaptic firing rate 5 Hz 40 Hz along with constant current 7.5 The variation in m,n and h are shown in blue, green and red. Pattern of action potential and firing rate: To quantitatively analyze the gating phenomenon, the spike pattern was observed (fig 7 and ). With the increase in the synaptic excitatory input, there is rapid opening of sodium channel and prolonged depolarization is restricted resulting in decrease in the amplitude of the action potential. Although, the firing rate of the modeled neuron is increasing with the increase in presynaptic excitatory firing rate. 2) Simulation results of model with synaptic excitatory input of pre-synaptic firing rate varying from 5 Hz to 40 Hz along with constant injected current of 7.5 μ /. We initially subjected the model to a range of increasing external current. Eventually, to determine the behavior of neuron to conductance based synaptic input, excitatory connections were activated. This excitatory input is poisson distributed with frequency varying from 5Hz to 40 Hz. a) Gating dynamics: As expected, the model responded in the same manner as in the earlier case. However, with the increase in the injected current the curves of n and h do not intersect each other. Signifying that opening of sodium channel is not affected by the increase in excitation but the inactivation of sodium channel and activation of potassium channel is partial and there is reduction in the resting period of sodium channel, this feature can be observed in fig.6 and. Fig.7 Voltage traces from the neuron model with synaptic excitatory input with presynaptic firing rate 5 Hz and 40 Hz along with constant current of 7.5. (c) Inter Spike Interval : The peculiar feature about the spike pattern is that the action potential generation is irregular in nature. This was quantified by the histogram of inter spike interval as shown in fig.8. This histogram reveals that the generation of action potential are of poisson distribution, which similar to as observed in biological system. Further, it was observed that with the increase in the presynaptic firing rate of the mean ISI of modeled neuron decreases as shown in fig

5 Similar observations are derived from the conductance curve (fig 10 and ). Furthermore, the comparison of the conductances led to a conclusion that sodium channels are surprising affected to a greater extent than potassium channel. 3) Simulation results of model with synaptic excitatory and inhibitory input of pre-synaptic firing rate varying form 5 Hz to 40 Hz along with constant injected current of 7.5µA/nF. Fig.8 Histogram of inter spike interval of a spiking neuron with synaptic input of presynaptic firing rate of 40 Hz along with a constant injected current of 7.5 Fig 9. Frequency current curve showing enhanced firing rate with increase in excitatory presynaptic input firing frequency (shown in blue). The red curve indicated the Inter spike interval, which reduces with increase in the firing rate. Next, we tried to understand if the above results could be reproduced with inhibitory input to the model. For this, we activated the inhibitory synaptic connections to the neuron. These inhibitory inputs were of variable presynaptic firing rate, which was subjected along with the presynaptic excitatory input and constant injected current of 7.5 µa nf. Neuron with both, excitatory and inhibitory inputs are more tractable to detailed biological neuron and allows for the precise identification of the mechanisms that underlie specific behavior. The model was excited with various ratios of excitatory and inhibitory inputs. Amongst the various ratios results of some are demonstrated here. Gating dynamics: We observed the same pattern of m, n and h for these inputs. As the ratio of excitatory to inhibitory presynaptic firing frequency increases, variation in opening of sodium channel is not observed but there is remarkable change in the inactivation of sodium channel and opening of potassium channel (fig 11 and). (d) Dynamics of sodium and potassium channel: We quantified the opening and closing of the gated channels with m, n, h curves and membrane potential curve. Fig 10. Sodium and Potassium conductances variation with synaptic input of presynaptic firing rate of 5 Hz and 40 Hz along with a constant injected current of 7.5.Red curve shows the sodium conductance and green shows potassium conductance. Fig 11.Variation of m, n, and h for excitatory to inhibitory presynaptic firing rate ratio 5:5 40:5. The variation in m,n and h are shown in blue, green and red. Fig. 12 suggests that with the ratio of 5:20 not much sodium channels are activated and there is a large resting period where as fig. 12 suggests 35:20 there a small almost negligible resting period. 1374

6 (c) Fig 12..Variation of m, n, and h for excitatory to inhibitory presynaptic firing rate ratio 5:20 35:20. The variation in m,n and h are shown in blue, green and red. Spike pattern and firing frequency and ISI: To understand how the spiking behavior changes with addition of inhibitory input, we fit the membrane potential curve. It reveals that spikes are irregular at these inputs. But with increase in excitation frequency the firing frequency of the neuron increases with a reduction in mean inter spike interval as is noticed from fig. 13,, (c) and (d). Further, fig. 13(e) depicts a relation between the firing rate of the modeled neuron and Inter spike interval of the generated action potential. With the increase in the presynaptic excitatory firing rate keeping the presynaptic inhibitory firing rate constant, it is asserted that the firing rate increases with increase in excitation and the mean inter spike interval decreases. Also, fig. 13 (f) and (g) represents the histogram of ISI which shows the generated action potential are poisson distributed. (d) (e) (f) (g) Fig.13 Illustrates the curve of membrane potential, neurons firing rate and inter spike interval histogram. voltage curve traced for the ratio 5:5, 40:5,(c)5:20 and (d)35:20. (e) Depicts the variation in the firing rate and ISI with increase in the excitatory pre synaptic firing rate keeping the inhibitory pre synaptic firing as constant. Blue curve is fit for the inhibitory presynaptic firing frequency of 5 Hz and red for 20 Hz. The yellow curve traces the value of mean ISI for inhibitory presynaptic frequency of 20 Hz and green curve for 5Hz. (f),(g) Represents ISI histogram for ratio 35:5 and 40:20 respectively. 1375

7 (c) Dynamics of sodium and potassium channel: Variation in sodium and potassium channels is also accepted. Because, as we are varying the ratio of presynaptic excitatory and inhibitory firing frequency the conductance amplitude is decreasing. In addition to this as visible from fig 14 -(d), at ratio 5:20 the sodium and potassium channel remains closed for a longer time. V. CONCLUSION Our modeling studies indicates that the fundamental mechanism that was responsible for generation of the action potential remains intact in Extended Hodgkin and Huxley model under various input conditions. That is the Hodgkin and Huxley model is robust, and the pattern of spike generation is same as that of original Hodgkin and Huxley model. This extended model gives a better realization to the biological neuron as a neuron with both excitatory and inhibitory inputs are more tractable to detailed biological neuron and allow precise identification of the mechanisms that underlie specific behavior. VI. REFERENCES [1] F.Rieke,D.Warland,R.de Ruyter van Steveninck, W.Bialek,Spikes- Exploring the Neural Code. Cambrige,MA:MIT Press,1997 [2] B.Cessac,T.Vieville,"On dynamics of Integrate-and -fire neuron network with conductane based syanpse".frontiers in Computaional Neuroscience 2[2],vol-2 july [3] L. F. Abbott," Lapicque s introduction of the integrate-and-fire model neuron (1907)",Brain Research Bulletin, Vol. 50, Nos. 5/6, pp , [4] C.Kirst, T.Marc, "How precise is the timing of action potential?", Frontier of Neuroscience vol-3, /neuro ,2009. [5]A.L. Hodgkin, A.F.Huxley; "A quantitative description of membrane current and its application to conduction and excitation in nerve"; J. Physiol.; 1952: 117, [6] A.Destexhe,M.Rudolph,D.Pare,"The high conductance state of neocortical neurons in vivo",nature Review Neurosciense, vol-4, /nrn1198, 2003 [7] R.Silver, Neuronal Arithmetic, Nature review, vol-11, /nrn 2864, [8] T.Vogels, L.F.Abott, "Gating multiple signals through detailed balance of excitation and inhibition in spiking networks",nature Neuroscience,vol- 12,no-4, /nn.2276,2009. [9] R.Wells,Biological Signal Processing, ch-3 (45-65). [10] E.R.Kandel,J.H.Schwartz,T.M.Jessell,Principles of Neuroscience. New York: McGraw-Hill,2000,ch-9. [11] T.Kispersky,J.Caplan,E.Marder,"Increase in sodium conductance decreases firing rate and gain in model neurons",j Neuroscience vol-32,no- 32;pg ,August (c) (d) Fig 14. Sodium and potassium conductance variation with the variation of firing frequency ratio 5:5, 40:5, (c) 5:20 and(d) 35:20. Red curve shows the sodium conductance and green shows potassium conductance. 1376